is Ms Grundgesetze der Arithmetik (The Fundamental Laws 01
Arithmetic), of which the first volume was published in 1893,
the second in 1903. In this work the concept of sets is used.
There is also a considerable use of more or less sarcastic invec-
tive against previous writers on the foundations of arithmetic
for their manifest blunders and manifold stupidities. The
second volume closes with the following acknowledgement.
A scientist can hardly encounter anything more unde-
sirable than to have the foundation collapse just as the
Trork is finished. I was put in this position by a letter from
Mr Bertrand Russell when the work was almost through
Russell had sent Frege his ingenious paradox of "the set of all
sets which are not members of themselves.' Is this set a member
of itself? Either answer can be puzzled out with a little thought
to be wrong. Yet Frege had freely used 'sets of all sets*.
Many ways were proposed for evading or ftltTninating the con-
tradictions which began exploding like a barrage in and over
the Frege-Dedekind-Cantor theory of the real numbers, con-
tinuity, and the infinite. Frege, Cantor, and Dedekind quit the
field, beaten and disheartened. Russell proposed his *vicious
circle principle' as a remedy: 'Whatever involves all of a collec-
tion must not be one of the collection'; later he put forth his
*asiom of redueibility% which, as it is now practically aban-
doned, need not be described. For a time these restoratives
were brilliantly effective (except in the opinion of the German
mathematicians, who never swallowed them). Gradually, as the
critical examination of all mathematical reasoning gained head-
way, physic was thrown to the dogs and a concerted effort was
begun to find out what really ailed the patient in his irrational
and real number system before administering further nostrums.
The present effort to understand our difficulties originated in
the work of David Hilbert (1862-1943) of Gottingen in 1609 and
in that of L. E. J. Brouwer (1881- ) of Amsterdam in 1012.
Both of these men and their numerous followers have the com-
mon purpose of putting mathematical reasoning on a sound
basis, although in several respects their methods and pnilo-